Silver ratio

For comparison, two quantities a, b with a > b > 0 are said to be in the golden ratio φ if,

${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi }$ 

However, they are in the silver ratio δ_S if,

${\displaystyle {\frac {2a+b}{a}}={\frac {a}{b}}=\delta _{S}.}$ 

Equivalently,

${\displaystyle 2+{\frac {b}{a}}={\frac {a}{b}}=\delta _{S}}$ 

Therefore,

${\displaystyle 2+{\frac {1}{\delta _{S}}}=\delta _{S}.}$ 

Multiplying by δ_S and rearranging gives

${\displaystyle {\delta _{S}}^{2}-2\delta _{S}-1=0.}$ 

Using the quadratic formula, two solutions can be obtained. Because δ_S is the ratio of positive quantities, it is necessarily positive, so,

${\displaystyle \delta _{S}=1+{\sqrt {2}}=2.41421356237\dots }$ 

The silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate 1 − √2 = ⁠−1/δ_S⁠ ≈ −0.41421 has absolute value less than 1. In fact it is the second smallest quadratic PV number after the golden ratio. This means the distance from δ ⁿ
_S to the nearest integer is ⁠1/δ ⁿ
_S⁠ ≈ 0.41421ⁿ. Thus, the sequence of fractional partsofδ ⁿ
_S, n = 1, 2, 3, ... (taken as elements of the torus) converges. In particular, this sequence is not equidistributed mod 1.

The lower powers of the silver ratio are

${\displaystyle \delta _{S}^{-1}=1\delta _{S}-2=[0;2,2,2,2,2,\dots ]\approx 0.41421}$ 

${\displaystyle \delta _{S}^{0}=0\delta _{S}+1=[$1]=1}  

${\displaystyle \delta _{S}^{1}=1\delta _{S}+0=[2;2,2,2,2,2,\dots ]\approx 2.41421}$ 

${\displaystyle \delta _{S}^{2}=2\delta _{S}+1=[5;1,4,1,4,1,\dots ]\approx 5.82842}$ 

${\displaystyle \delta _{S}^{3}=5\delta _{S}+2=[14;14,14,14,\dots ]\approx 14.07107}$ 

${\displaystyle \delta _{S}^{4}=12\delta _{S}+5=[33;1,32,1,32,\dots ]\approx 33.97056}$ 

The powers continue in the pattern

${\displaystyle \delta _{S}^{n}=K_{n}\delta _{S}+K_{n-1}}$ 

where

${\displaystyle K_{n}=2K_{n-1}+K_{n-2}}$ 

For example, using this property:

${\displaystyle \delta _{S}^{5}=29\delta _{S}+12=[82;82,82,82,\dots ]\approx 82.01219}$ 

Using K₀ = 1 and K₁ = 2 as initial conditions, a Binet-like formula results from solving the recurrence relation

${\displaystyle K_{n}=2K_{n-1}+K_{n-2}}$ 

which becomes

${\displaystyle K_{n}={\frac {1}{2{\sqrt {2}}}}\left(\delta _{S}^{n+1}-{(2-\delta _{S})}^{n+1}\right)}$ 

The silver ratio is intimately connected to trigonometric ratios for ⁠π/8⁠ = 22.5°.

${\displaystyle \tan {\frac {\pi }{8}}={\sqrt {2}}-1={\frac {1}{\delta _{s}}}}$ 

${\displaystyle \cot {\frac {\pi }{8}}=\tan {\frac {3\pi }{8}}={\sqrt {2}}+1=\delta _{s}}$ 

So the area of a regular octagon with side length a is given by

${\displaystyle A=2a^{2}\cot {\frac {\pi }{8}}=2\delta _{s}a^{2}\simeq 4.828427a^{2}.}$ 

• Metallic means
• Golden ratio
• Supersilver ratio
• Ammann–Beenker tiling

• Buitrago, Antonia Redondo (2008). "Polygons, Diagonals, and the Bronze Mean", Nexus Network Journal 9,2: Architecture and Mathematics, p.321-2. Springer Science & Business Media. ISBN 9783764386993.

• Weisstein, Eric W. "Silver Ratio". MathWorld.
• "An Introduction to Continued Fractions: The Silver Means Archived 2018-12-08 at the Wayback Machine", Fibonacci Numbers and the Golden Section.
• "Silver rectangle and its sequence" at Tartapelago by Giorgio Pietrocola